Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 41 }$ to precision 11.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 41 }$: $ x^{3} + x + 35 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 20 a^{2} + 11 a + 6 + \left(24 a^{2} + 5 a + 37\right)\cdot 41 + \left(38 a^{2} + 34 a + 28\right)\cdot 41^{2} + \left(13 a^{2} + 34 a + 37\right)\cdot 41^{3} + \left(16 a^{2} + 12 a + 23\right)\cdot 41^{4} + \left(6 a^{2} + 5 a + 2\right)\cdot 41^{5} + \left(25 a^{2} + 27 a + 34\right)\cdot 41^{6} + \left(32 a^{2} + 15 a + 38\right)\cdot 41^{7} + \left(7 a^{2} + 16 a + 15\right)\cdot 41^{8} + \left(6 a^{2} + 12 a + 19\right)\cdot 41^{9} + \left(12 a^{2} + 40 a + 1\right)\cdot 41^{10} +O\left(41^{ 11 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 39 + 15\cdot 41 + 31\cdot 41^{2} + 23\cdot 41^{3} + 30\cdot 41^{4} + 22\cdot 41^{6} + 8\cdot 41^{7} + 22\cdot 41^{8} + 34\cdot 41^{9} + 6\cdot 41^{10} +O\left(41^{ 11 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 10 + 31\cdot 41 + 33\cdot 41^{2} + 20\cdot 41^{3} + 35\cdot 41^{4} + 14\cdot 41^{5} + 37\cdot 41^{6} + 36\cdot 41^{7} + 14\cdot 41^{8} + 39\cdot 41^{9} + 11\cdot 41^{10} +O\left(41^{ 11 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 23 a^{2} + 31 a + 8 + \left(31 a^{2} + 16 a + 28\right)\cdot 41 + \left(3 a^{2} + 15 a + 5\right)\cdot 41^{2} + \left(21 a^{2} + 8 a + 15\right)\cdot 41^{3} + \left(28 a + 13\right)\cdot 41^{4} + \left(3 a^{2} + 20 a\right)\cdot 41^{5} + \left(23 a^{2} + 14 a + 19\right)\cdot 41^{6} + \left(21 a^{2} + 7 a + 31\right)\cdot 41^{7} + \left(32 a^{2} + 27 a + 18\right)\cdot 41^{8} + \left(32 a^{2} + 3 a + 23\right)\cdot 41^{9} + \left(7 a^{2} + 23 a + 39\right)\cdot 41^{10} +O\left(41^{ 11 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 9 a^{2} + 24 a + 40 + \left(17 a^{2} + 31 a + 1\right)\cdot 41 + \left(16 a^{2} + 6 a + 12\right)\cdot 41^{2} + \left(36 a^{2} + 32 a + 23\right)\cdot 41^{3} + \left(19 a^{2} + 40 a + 39\right)\cdot 41^{4} + \left(16 a^{2} + 4 a + 24\right)\cdot 41^{5} + \left(7 a^{2} + 7 a + 10\right)\cdot 41^{6} + \left(36 a^{2} + 9 a + 39\right)\cdot 41^{7} + \left(4 a^{2} + 22 a + 2\right)\cdot 41^{8} + \left(14 a^{2} + 15 a + 40\right)\cdot 41^{9} + \left(33 a^{2} + 32 a + 21\right)\cdot 41^{10} +O\left(41^{ 11 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 37 + 4\cdot 41^{2} + 37\cdot 41^{3} + 20\cdot 41^{4} + 29\cdot 41^{5} + 35\cdot 41^{6} + 21\cdot 41^{7} + 13\cdot 41^{8} + 34\cdot 41^{9} + 41^{10} +O\left(41^{ 11 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 39 a^{2} + 40 a + 5 + \left(25 a^{2} + 18 a + 38\right)\cdot 41 + \left(39 a^{2} + 32 a + 15\right)\cdot 41^{2} + \left(5 a^{2} + 38 a + 32\right)\cdot 41^{3} + \left(24 a^{2} + 40 a + 1\right)\cdot 41^{4} + \left(31 a^{2} + 14 a + 33\right)\cdot 41^{5} + \left(33 a^{2} + 40 a + 39\right)\cdot 41^{6} + \left(27 a^{2} + 17 a + 21\right)\cdot 41^{7} + \left(38 a + 38\right)\cdot 41^{8} + \left(2 a^{2} + 24 a + 2\right)\cdot 41^{9} + \left(21 a^{2} + 18 a + 21\right)\cdot 41^{10} +O\left(41^{ 11 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 17 a^{2} + 23 a + 18 + \left(30 a^{2} + 14 a + 24\right)\cdot 41 + \left(34 a^{2} + 31 a + 10\right)\cdot 41^{2} + \left(40 a^{2} + 18 a + 26\right)\cdot 41^{3} + \left(9 a^{2} + 6 a + 5\right)\cdot 41^{4} + \left(28 a^{2} + 38 a + 19\right)\cdot 41^{5} + \left(3 a^{2} + 3 a + 35\right)\cdot 41^{6} + \left(a^{2} + 21 a + 15\right)\cdot 41^{7} + \left(9 a^{2} + 36 a + 19\right)\cdot 41^{8} + \left(26 a^{2} + 33 a + 34\right)\cdot 41^{9} + \left(34 a^{2} + a + 22\right)\cdot 41^{10} +O\left(41^{ 11 }\right)$ |
| $r_{ 9 }$ |
$=$ |
$ 15 a^{2} + 35 a + 3 + \left(34 a^{2} + 35 a + 27\right)\cdot 41 + \left(30 a^{2} + 2 a + 21\right)\cdot 41^{2} + \left(4 a^{2} + 31 a + 29\right)\cdot 41^{3} + \left(11 a^{2} + 34 a + 33\right)\cdot 41^{4} + \left(37 a^{2} + 38 a + 38\right)\cdot 41^{5} + \left(29 a^{2} + 29 a + 11\right)\cdot 41^{6} + \left(3 a^{2} + 10 a + 31\right)\cdot 41^{7} + \left(27 a^{2} + 23 a + 17\right)\cdot 41^{8} + \left(32 a + 17\right)\cdot 41^{9} + \left(14 a^{2} + 6 a + 36\right)\cdot 41^{10} +O\left(41^{ 11 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 9 }$
| Cycle notation |
| $(2,8)(3,5)(6,9)$ |
| $(1,8,2)(3,7,5)(4,9,6)$ |
| $(1,7,4)(2,3,6)(5,9,8)$ |
| $(2,6,3)(5,9,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 9 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $9$ | $2$ | $(2,8)(3,5)(6,9)$ | $1$ |
| $1$ | $3$ | $(1,7,4)(2,3,6)(5,9,8)$ | $3 \zeta_{3}$ |
| $1$ | $3$ | $(1,4,7)(2,6,3)(5,8,9)$ | $-3 \zeta_{3} - 3$ |
| $6$ | $3$ | $(1,8,2)(3,7,5)(4,9,6)$ | $0$ |
| $6$ | $3$ | $(1,5,2)(3,7,9)(4,8,6)$ | $0$ |
| $6$ | $3$ | $(1,9,2)(3,7,8)(4,5,6)$ | $0$ |
| $6$ | $3$ | $(2,6,3)(5,9,8)$ | $0$ |
| $9$ | $6$ | $(1,7,4)(2,5,6,8,3,9)$ | $\zeta_{3}$ |
| $9$ | $6$ | $(1,4,7)(2,9,3,8,6,5)$ | $-\zeta_{3} - 1$ |
The blue line marks the conjugacy class containing complex conjugation.
